Jim

Daikon radish
-This is a staple-- you can use this in ANYTHING, from salads to stews to pickles. A common table condiment is grated radish mixed with soy sauce which you then mix into your rice. It's a great sore throat remedy. Also add grated radish to tempura dipping sauce (if you eat at a Japanese restaurant request for them to bring it out for you ;3)

Kyuri (Japanese cucumber)
-This is the best tasting cucumber IN. THE. WORLD. Slender, dark green, bumpy, and crispy, it's the only cucumber you will ever want to eat. Ever. Ever. I love them fresh or pickled. They make such wonderful pickles.

Satsuma imo (Satsuma sweet potato)
-Sweet doesn't even begin to describe this potato. It's not cloying like the more orangey ones are, but rather subtle but still very flavorful with a creamy consistency. My family likes to steam or boil it and put salt in the holes where they dug out the eyes (but then again they also put salt on apples and watermelons so... yeah). I like it au naturel. It's very pretty-- purple on the outside, cream-colored on the inside.

Murasaki imo (Purple sweet potato)
-Truthfully, I have never eaten this except as prepared in fancy desserts, like cakes and ice creams. I will assume that it's sweet, but I know for a fact that the flesh is very very purple. If nothing else, it's an extremely beautiful potato.

Shiso (Japanese... basil? or beefsteak plant)
-There are two kinds-- red and green. They have a sort of basily smell, but not quite? The taste is sort of spicy/tangy/bitter/basil. Anyway, they are delicious fresh and pickled, as garnishes, or just a pretty ornamental plant. The red shiso is particularly beautiful.

Kabocha (Japanese pumpkin)
-This is probably my favorite of all Japanese vegetables. It's a smallish, green/orange bumpy pumpkin with the most delicious pumpkin taste ever. It's sweet, it's creamy, it's bright orange and cheerful. It will make you happy when it gets dark during the fall and winter.

Negi (spring onion)
-This ain't your ordinary green onion-- in Japan, these things are enormous-- like, 2+ feet long enormous. Even though they taste pretty much like most green onions, you don't have to do as much slicing because one of these will equal a couple bunches of the small ones.

Gobou (burdock root)
-I remember hearing a sad story where one of the pieces of evidence used against the Japanese for their war crimes was that they forced American POWs to eat yucky tree roots instead of real food while in the camps. I'm glad that war criminals got prosecuted for what they did, but bringing the humble gobou plant into this was unfair-- those yucky roots are a favorite of macrobiotic cooks today because they're hella good for you, and delicious to boot :9

Nasu (Japanese eggplant)
-Long and slender and dark in color, these guys pack such a wonderful flavor in such a skinny package. They're great as pickles and tempura, but there's a wide array of dishes you can use them in.

Then there's the leafy and green vegetables, of which there are many that are widely available-- napa cabbage, mizuna, mibuna, gailan, bok choy, etc. Note that Japanese napa cabbage (hakusai) is ENORMOUS compared to what you buy in grocery stores here-- I am not sure if they just grow it longer or if it's a different variety.

I also enjoy Indian cooking, and popular veggies used in the various regional cuisines include okra, snap beans, small round eggplants, cauliflower, lentils, potatoes, and tomatoes. Just remember to grow all the right spices to throw in you dishes ;3

I can keep going on and on but I'll spare you. I think you have plenty of ideas going on in this thread. Especially if your daughter loves Asian foods, she'll have an easy time being a vegetarian. Who knows-- maybe you can start growing soybeans so you can make your own tofu :D